Fifty rupee note = 3
Hundred rupee note = 4
500 rupee note = 6
Totsl = 13
\(\mathrm{n}(\mathrm{S})=13 \mathrm{C}_{2}=78\)
\(\text {No. of ways the event to occur }=4\)
\(n(A)=4 C_{2}=6\)
\(\text {No. of ways the event not to occur }=72\)
The odds of event are a:b
\(6: 72 \Rightarrow 1: 12\)

\(\int 2^{3 x+5} d x =\frac{2^{3 x+5}}{\log 2 \times \frac{d}{d x}(3 x+5)}+c \)
\(=\frac{2^{3 x+5}}{3 \log 2}+c \)

\(x =a \sin \theta, y=b \cos \theta \)
\(\frac{d x}{d \theta} =a \cos \theta, \frac{d y}{d \theta}=-b \sin \theta\)
\(\frac{d y}{d x} =\frac{d y / d \theta}{d x / d \theta}=\frac{-b \sin \theta}{a \cos \theta}=\frac{-b}{a} \tan \theta \)
\(\frac{d^{2} y}{d x^{2}} =\frac{-b}{a} \sec ^{2} \theta \frac{d \theta}{d x}\)
\(=\frac{-b}{a} \sec ^{2} \theta \cdot \frac{1}{a \cos \theta} \)
\(= \frac{-b}{a^{2}} \sec ^{3} \theta \)

\(f(x)=x(-1)^{\left\lfloor\frac{1}{x}\right\rfloor}\)
\(=x(\pm 1) \left(\because\left[\frac{1}{x}\right\rfloor \text { is an integer }\right) \)
\(=\pm x \)
\(\therefore \lim _{x \rightarrow 0} f(x)=\lim _{x \rightarrow 0}(\pm x)=0\)

\(2 \vec{a}=\vec{b}+\vec{c} \Rightarrow \vec{a}+\vec{a}=\vec{b}+\vec{c} \Rightarrow \vec{a}-\vec{b}=\vec{c}-\vec{a} \)
\(\overrightarrow{O A}-\overrightarrow{O B}=\overrightarrow{O C}-\overrightarrow{O A} \Rightarrow \overrightarrow{B A}=\overrightarrow{A C} \)
\(\Rightarrow \vec{a}, \vec{b}, \vec{c} \text { are collinear }\)

\(A A^{T}=9 I \)
\({\left[\begin{array}{ccc} 1 & 2 & 2 \\ 2 & 1 & -2 \\ a & 2 & b \end{array}\right]\left[\begin{array}{ccc} 1 & 2 & a \\ 2 & 1 & 2 \\ 2 & -2 & b \end{array}\right]=\left[\begin{array}{ccc} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{array}\right]} \)
\({\left[\begin{array}{ccc} 1+4+4 & 2+2-4 & a+4+2 b \\ 2+2-4 & 4+1+4 & 2 a+2-2 b \\ a+4+2 b & 2 a+2-2 b & a^{2}+4+b^{2} \end{array}\right]} \)
\(=\left[\begin{array}{ccc} 9 & 0 & 0 \\ 0 & 9 & 0 \\ 0 & 0 & 9 \end{array}\right] \)
\(a+4+2 b =0 \)
\(a+2 b =-4 \)
\(2 a-2 b+2 =0 \)
\(2 a-2 b =-2 \)
\(\text { Solve (1) } \&(2) \text { we get } a=-2 b=-1\)
\(\therefore(a, b)=(-2,1)\)

\(\text {Perimeter }=4+2 \sqrt{2}\)
\(a+a+\sqrt{2} =4+2 \sqrt{2} \)
\(2 a+\sqrt{2} a =4+2 \sqrt{2} \)
\(\therefore a =2 \)
\(\text {Equation of line is } \frac{x}{2}+\frac{y}{2}=1\)
\(x+y-2=0\)

The number of five digit telephone numbers which can be formed using the digits 0,1,2.... 9 is 10⁵.
The number of 5 digit numbers which has none of their digit repeated is ¹⁰P₅, = 30240
The required number of telephone. number is 10⁵ =- 30240 = 69,760

\(\log _{\sqrt{x}} 0.25 =4 \)
\((\sqrt{x})^{4} =0.25 \)
\((\sqrt{x})^{4} =\frac{1}{4} \)
\(x^{2} =\frac{1}{4} \)
\(\Rightarrow x=\frac{1}{2}=0.5\)

\(\frac{\sin (A-B)}{\cos A \cos B} =\frac{\sin A \cos B-\cos A \sin B}{\cos A \cos B} \)
\(=\tan A-\tan B \)
\(\text { L.H.S } =\tan A-\tan B+\tan B-\tan C+\tan C-\tan A=0\)